Richard Lenz

Luciano Melodia, Richard Lenz

In this study the Voronoi interpolation is used to interpolate a set of points drawn from a topological space with higher homology groups on its filtration. The technique is based on Voronoi tessellation, which induces a natural dual map to the Delaunay triangulation. Advantage is taken from this fact calculating the persistent homology on it after each iteration to capture the changing topology of the data. The boundary points are identified as critical. The Bottleneck and Wasserstein distance serve as a measure of quality between the original point set and the interpolation. If the norm of two distances exceeds a heuristically determined threshold, the algorithm terminates. We give the theoretical basis for this approach and justify its validity with numerical experiments.

Luciano Melodia, Richard Lenz

In this paper we present an approach to determine the smallest possible number of neurons in a layer of a neural network in such a way that the topology of the input space can be learned sufficiently well. We introduce a general procedure based on persistent homology to investigate topological invariants of the manifold on which we suspect the data set. We specify the required dimensions precisely, assuming that there is a smooth manifold on or near which the data are located. Furthermore, we require that this space is connected and has a commutative group structure in the mathematical sense. These assumptions allow us to derive a decomposition of the underlying space whose topology is well known. We use the representatives of the $k$-dimensional homology groups from the persistence landscape to determine an integer dimension for this decomposition. This number is the dimension of the embedding that is capable of capturing the topology of the data manifold. We derive the theory and validate it experimentally on toy data sets.

Luciano Melodia, Richard Lenz

In this paper, we use topological data analysis techniques to construct a suitable neural network classifier for the task of learning sensor signals of entire power plants according to their reference designation system. We use representations of persistence diagrams to derive necessary preprocessing steps and visualize the large amounts of data. We derive deep architectures with one-dimensional convolutional layers combined with stacked long short-term memories as residual networks suitable for processing the persistence features. We combine three separate sub-networks, obtaining as input the time series itself and a representation of the persistent homology for the zeroth and first dimension. We give a mathematical derivation for most of the used hyper-parameters. For validation, numerical experiments were performed with sensor data from four power plants of the same construction type.

Luciano Melodia, Richard Lenz

In this paper we present an approach to determine the smallest possible number of neurons in a layer of a neural network in such a way that the topology of the input space can be learned sufficiently well. We introduce a general procedure based on persistent homology to investigate topological invariants of the manifold on which we suspect the data set. We specify the required dimensions precisely, assuming that there is a smooth manifold on or near which the data are located. Furthermore, we require that this space is connected and has a commutative group structure in the mathematical sense. These assumptions allow us to derive a decomposition of the underlying space whose topology is well known. We use the representatives of the $k$-dimensional homology groups from the persistence landscape to determine an integer dimension for this decomposition. This number is the dimension of the embedding that is capable of capturing the topology of the data manifold. We derive the theory and validate it experimentally on toy data sets.

Luciano Melodia, Richard Lenz

In this study the Voronoi interpolation is used to interpolate a set of points drawn from a topological space with higher homology groups on its filtration. The technique is based on Voronoi tessellation, which induces a natural dual map to the Delaunay triangulation. Advantage is taken from this fact calculating the persistent homology on it after each iteration to capture the changing topology of the data. The boundary points are identified as critical. The Bottleneck and Wasserstein distance serve as a measure of quality between the original point set and the interpolation. If the norm of two distances exceeds a heuristically determined threshold, the algorithm terminates. We give the theoretical basis for this approach and justify its validity with numerical experiments.

Andreas M. Wahl, Gregor Endler, Peter K. Schwab et al.

In larger organizations, multiple teams of data scientists have to integrate data from heterogeneous data sources as preparation for data analysis tasks. Writing effective analytical queries requires data scientists to have in-depth knowledge of the existence, semantics, and usage context of data sources. Once gathered, such knowledge is informally shared within a specific team of data scientists, but usually is neither formalized nor shared with other teams. Potential synergies remain unused. We therefore introduce a novel approach which extends data management systems with additional knowledge-sharing capabilities to facilitate user collaboration without altering established data analysis workflows. Relevant collective knowledge from the query log is extracted to support data source discovery and incremental data integration. Extracted knowledge is formalized and provided at query time.